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I was attempting to but several large files from my local hard disk and paste them into my external harddrive. It failed (nothing restarted or anything, but perhaps I forgot to unmount properly I don't know) and now I can't find the files anywhere.

Is there a particular location that I should look where such files might be temporarily stored?

They don't actually go anywhere. The filing system action isn't started until you 'paste'. Then they're copied over and after that if the starting action wasn't a 'copy' but a 'cut', the original is deleted (a 'cut / paste' on the same volume is different, but that's not relevant here). If the copy action fails and the filing system notices this, the delete action is skipped.

You write 'It failed'. How exactly? Did you get an error or did it fail silently, i.e. did you only notice the files weren't on the external drive (much) later? In that case I would think you have indeed forgotten to unmount. The files will still be on the external drive, but the mapping hasn't been updated, so they are inaccessible. Maybe file recovery software can help you get them back.

Good question. They have to be somewhere even temporarily. I tried cut and pasting a video from my laptop to external drive and it was not successful and I lost it completely. So good question. Since then I've always done "copy", paste then manual delete.